C2 positivity-preserving rational interpolation splines in one and two dimensions

A class of rational quartic/cubic interpolation spline with two local control parameters is presented, which can be C-2 continuous without solving a linear system of consistency equations for the derivative values at the knots. The effects of the local control parameters on generating interpolation curves are illustrated. For C-2 interpolation, the given interpolant can locally reproduce quadratic polynomials and has O (h(2)) or O (h(3)) convergence. Simple schemes for the C-2 interpolant to preserve the shape of 2D positive data are developed. Moreover, based on the Boolean sum of quintic interpolating operators, a class of bi-quintic partially blended rational quartic/ cubic interpolation surfaces is also constructed. The given interpolation surface provides four local control parameters and can be C-2 continuous without using the second or higher mixed partial derivatives on a rectangular grid. Simple sufficient data dependent constraints are also derived on the local control parameters to preserve the shape of a 3D positive data set arranged over a rectangular grid. (C) 2017 Elsevier Inc. All rights reserved.